Any measurable function into (ℝ,ℬ⁢(ℝ)), where ℬ⁢(ℝ) is the Borel sigma algebra of the real numbersℝ, is called a Borel measurable function.11More generally, a measurable function is called Borel measurable if the range space Y is a topological space with ℬ⁢(Y) the sigma algebra generated by all open sets of Y. The space of all Borel measurable functions from a measurable space (X,ℬ⁢(X)) is denoted by ℒ0⁢(X,ℬ⁢(X)).

Similarly, we write ℒ¯0⁢(X,ℬ⁢(X)) for ℳ⁢((X,ℬ⁢(X)),(ℝ¯,ℬ⁢(ℝ¯))), where ℬ⁢(ℝ¯) is the Borel sigma algebra of ℝ¯, the set of extended real numbers.

Remark. If f:X→Y and g:Y→Z are measurable functions, then so is g∘f:X→Z, for if E is ℬ⁢(Z)-measurable, then g-1⁢(E) is ℬ⁢(Y)-measurable, and f-1⁢(g-1⁢(E)) is ℬ⁢(X)-measurable. But f-1⁢(g-1⁢(E))=(g∘f)-1⁢(E), which implies that g∘f is a measurable function.

Example:

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Let E be a subset of a measurable space X. Then the characteristic functionχE is a measurable function if and only if E is measurable.